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SageMath
E = EllipticCurve("ci1")
E.isogeny_class()
Elliptic curves in class 30912.ci
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 30912.ci1 | 30912y6 | \([0, 1, 0, -5064577, -4388596513]\) | \(54804145548726848737/637608031452\) | \(167145119796953088\) | \([2]\) | \(786432\) | \(2.4557\) | |
| 30912.ci2 | 30912y4 | \([0, 1, 0, -1133697, 464225247]\) | \(614716917569296417/19093020912\) | \(5005120873955328\) | \([2]\) | \(393216\) | \(2.1091\) | |
| 30912.ci3 | 30912y3 | \([0, 1, 0, -324737, -64914465]\) | \(14447092394873377/1439452851984\) | \(377343928430493696\) | \([2, 2]\) | \(393216\) | \(2.1091\) | |
| 30912.ci4 | 30912y2 | \([0, 1, 0, -73857, 6586335]\) | \(169967019783457/26337394944\) | \(6904190060199936\) | \([2, 2]\) | \(196608\) | \(1.7625\) | |
| 30912.ci5 | 30912y1 | \([0, 1, 0, 8063, 573407]\) | \(221115865823/664731648\) | \(-174255413133312\) | \([2]\) | \(98304\) | \(1.4160\) | \(\Gamma_0(N)\)-optimal |
| 30912.ci6 | 30912y5 | \([0, 1, 0, 401023, -313269537]\) | \(27207619911317663/177609314617308\) | \(-46559216171039588352\) | \([2]\) | \(786432\) | \(2.4557\) |
Rank
sage: E.rank()
The elliptic curves in class 30912.ci have rank \(1\).
Complex multiplication
The elliptic curves in class 30912.ci do not have complex multiplication.Modular form 30912.2.a.ci
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
